We first provide an approach to the recent conjecture of Bierstone-Milman-Pawlucki on Whitney's old problem on smooth extendability of functions defined on a closed subset of a Euclidean space, using higher order paratangent bundle they introduced. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek's theorem on controllability of flatness by the values on a closed set. The multi-dimensional Vandermonde matrix plays an important role in both cases.
We prove Fourier restriction estimates to arbitrary compact $C^{N}$ curves for any $N > d$ in the (sharp) Drury range, using a power of the affine arclength measure as a mitigating factor. In particular, we make no nondegeneracy assumption on the curve.
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.
Recently Wolff obtained a sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of ``elliptic surfaces'' such as paraboloids and spheres. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.
We improve the Bennett--Carbery--Tao trilinear restriction estimate for subsets of the paraboloid in three dimensions, giving the sharp factor depending on the transversality.
We show that there are functions $f$ in the Hölder class $C^{ { α}}[0,1]$, $1< { α}<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \subset } [0,1]$ with $ { \bar { dim }_M } A> { α}-1$. Our earlier result shows that for the typical/generic $f\in { C_ { 1 } ^ { { α} } [0,1] }$, $0\leq { α}<2$ there is always a set $A { \subset } [0,1]$ such that $f|_A$ is convex and $ { \bar { dim }_M } A=1$. The analogous statement for monotone restrictions is the following: there are functions $f$ in the Hölder class $C^{ { α}}[0,1]$, $1/2 \leq { α}<1$ such that $f|_{A}$ is not monotone on $A {…
Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp $L^p$ null form estimates.
Given a continuous real-valued function on [0, 1], and a closed subset E \subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The restriction f E will typically be "better behaved" than f . It may have bounded variation when f doesn't, it may have a better modulus of continuity than f, it may be monotone when f is not, etc. All this clearly depends on f and on E, and the questions that we discuss here are about the existence, for every f, or every f in some class, of "substantial" sets E such t…
We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maxi…
Suppose $0 < α\leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_α(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^α$ holds for all balls $B \subset \Bbb R^n$ of radius $R \geq 1$. After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form $\| Ef \|_{L^p(B,Hdx)} \leq C R^εA_α(H)^{1/p} \| f \|_{L^q(σ)}$ have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, …
We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th moment of (possibly weighted) sums, for any real $p > 2$ and $\log^{C(p)}x \leq y \leq x$. Our proof develops an argument of Bourgain, showing this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our boun…
Suppose that $f$ belongs to a suitably defined complete metric space $ {\cal C}^{α}$ of Hölder $ α$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A {\subset} [0,1]$ such that $f|_{A}$ is monotone, or convex/concave. Some of our results are about generic functions in $ {\cal C}^{α}$ like the following one: we prove that for the generic $f\in C_{1}^{α}[0,1]$, $0\leq α<2$ for any $A {\subset} [0,1]$ such that $f|_{A}$ is convex, or concave we have ${\mathrm{dim}}_{\mathrm H} A\leq \underline{\mat…
Let $ϕ$ be a smooth function on a compact interval $I$. Let $$γ(t)=\left (t,t^2,\cdots,t^{n-1},ϕ(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(γ(t))\big|^q \big|ϕ^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$ holds in the range $$1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sjölin (1974) for $n=2$. Our proof relies on ideas of Sjölin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.
Let $S$ be a hypersurface in $\Bbb R^3$ which is the graph of a smooth, finite type function $ϕ,$ and let $μ=ρ\, d\si$ be a surface carried measure on $S,$ where $d\si$ denotes the surface element on $S$ and $ρ$ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform $\hat μ$ of $μ,$ which are sharp except for the case where the principal face of the Newton polyhedron of $ϕ,$ when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp $L^p$-$L^2$ Fourier restriction theorem for $S$ in the case where the original coordin…
Restrictions of Békollé--Bonami weights and Bloch functions
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Umeå University（于默奥大学）；Universidad Autónoma de Madrid（马德里自治大学）；Norwegian University of Science and Technology（挪威科技大学）
If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\| E_S f\|_{L^p(\mathbb{R}^3)} \le C(p,S) \| f \|_{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.
The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general $L^p(R^3)\to L^q(S)$ Fourier restriction estimates, by studying a prototypical class of two-dimensional surfaces with strongly varying curvature conditions. Our approach is based on an adaptation of the so-called bilinear method. We discuss several new features arising in the study of this problem.
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area. The restriction problem are connected to many other conjectures, notably the Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as t…
It is shown that an arbitrary function from $D\subset \R^n$ to $\R^m$ will become $C^{0,α}$-continuous in almost every $x\in D$ after restriction to a certain subset with limit point $x$. For $n\geq m$ differentiability can be obtained. Examples show the Hölder exponent $α=\min\{1,\frac{n}{m}\}$ is optimal.
For $0<α<1$ let $V(α)$ denote the supremum of the numbers $v$ such that every $α$-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \leq V(α)\leq 1/(2-α)$ and asked whether the upper bound is sharp. We show that in fact $V(α)=\max\{1/2,α\}$. Let $\dim_{H}$ and $\overline{\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(α)$ is a consequence of the following theorem. Let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. Then, almos…
This is the first of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in $\Bbb R^3$, which includes in particular all real-analytic hypersurfaces. The present file is a modified and extended version of an earlier file of the same title. Some changes and corrections had become necessary, in order to make sure that Part II is really compatible with Part I, and it is recommended to replace the earlier version of Part I by the new one.
Given $A \subset F_{p}^2$ a sufficiently small set in the plane over a prime residue field, we prove that there are at most $O_ε(|A|^{\frac{99}{41}+ε})$ rectangles with corners in $A$. The exponent $\frac{99}{41} = 2.413\ldots$ improves slightly on the exponent of $\frac{17}{7} = 2.428\ldots$ due to Rudnev and Shkredov. Using this estimate we prove that the extension operator for the three dimensional paraboloid in prime order fields maps $L^2 \rightarrow L^{r}$ for $r >\frac{188}{53}=3.547\ldots$ improving the previous range of $r\geq \frac{32}{9}= 3.\overline{555}$.
